325_PracticePS8_Soln

325_PracticePS8_Soln - Department of Economics University...

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Department of Economics University of Maryland Economics 325 Intermediate Macroeconomic Analysis Practice Problem Set 8 Suggested Solutions Professor Sanjay Chugh Spring 2011 1. The Yield Curve. An important indicator of markets’ beliefs/expectations about the future path of the macroeconomy is the “yield curve,” which, simply put, describes the relationship between the maturity length of a particular bond (recall that bonds come in various maturity lengths) and the per-year interest rate on that bond. A bond’s “yield” is alternative terminology for its interest rate. A sample yield curve is shown in the following diagram: This diagram plots the yield curve for U.S. Treasury bonds that existed in markets on February 9, 2005: as it shows, a 5-year Treasury bond on that date carried an interest rate of about 4 percent, a 10-year Treasury bond on that date carried an interest rate of about 4.4 percent, and a 30-year Treasury bond on that date carried an interest rate of about 4.52 percent. Recall from our study of bond markets that prices of bonds and nominal interest rates on bonds are negatively related to each other. The yield curve is typically discussed in terms of nominal interest rates (as in the graph above). However, because of the inverse

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2 relationship between interest rates on bonds and prices of bonds, the yield curve could equivalently be discussed in terms of the prices of bonds. In this problem, you will use an enriched version of our infinite-period monetary framework from Chapter 14 to study how the yield curve is determined. Specifically, rather than assuming the representative consumer has only one type of bond (a one- period bond) he can purchase, we will assume the representative consumer has several types of bonds he can purchase – a one-period bond, a two-period bond, and, in the later parts of the problem, a three-period bond. Let’s start just with two-period bonds. We will model the two-period bond in the simplest possible way: in period t , the consumer purchases TWO t B units of two-period bonds, each of which has a market price , b TWO t P and a face value of one (i.e., when the two-period bond pays off, it pays back one dollar). The defining feature of a two- period bond is that it pays back its face value two periods after purchase (indeed, hence the term “two-period bond”…). The one-period bond is just as we have discussed in class and in Chapter 14. Mathematically, then, suppose (just as in Chapter 14) that the representative consumer has a lifetime utility function starting from period t 2 2 3 3 1 2 3 1 2 3 1 2 3 ln ln ln ln ln ln ln ln ... t t t t t t t t t t t t M M M M c c c c P P P P E E E E E E ± ± ± ± ± ± ± ± ± § · § · § · § · ± ± ± ± ± ± ± ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ © ¹ © ¹ © ¹ , and his period- t budget constraint is given by , 1 1 2 1 ( ) b b TWO TWO TWO t t t t t t t t t t t t t t t t Pc P B P B M S a Y M B B S D a ² ² ² ² ± ± ± ± ± ± ± ± ± .
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